Finite Group Databasebeta

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Cayley diagram

D5×#126 #66037

order320
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C24

Aliases

direct productD5×#126

Splittings

FittingC5×#126 ⋊ C2
SylowC5 ⋊ C2×#126

Subgroups

subgroupG/H
center, FrattiniC22C23×D5
commutator, socle, cosocleC22×C5C24
FittingC5×#126C2
prime radicalC23×D5C22
hypercenter#126D5
hypocenterC5C2×#126
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ #126

246 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 4 generators on 21 points:

(1,2,4,5,3)(7,11)(10,13)(14,16,17,21)(15,19,18,20);(6,8,9,12)(7,10,11,13)(15,18)(19,20);(6,7)(8,10)(9,11)(12,13)(14,15)(16,20)(17,18)(19,21);(2,3)(4,5)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×#126, #10617358, #10617359, …
maximal normal subgroup ofAGL(1,𝔽5)×#126, #10617358, #10617359, …
maximal quotient of#10605990, #10605997, #10605998, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1375);